Exponential sums involving Maass forms

We study the exponential sums involving Fourier coefficients of Maass forms and exponential functions of the form e(αn^β),where 0≠α∈? and 0<β<1. An asymptotic formula is proved for the nonlinear exponential sum ∑X<n≤2Xλg(n)e(αnβ), when β = 1/2 and |α| is close to 2q, q∈?+, where λ_g(n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL₂(?). The similar natures of the divisor function τ(n) and the representation function r(n) in the circle problem in nonlinear exponential sums of the above type are also studied.